Norman Allan
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Poincare's Recurrence

 

 

 
     
  If certain kinds of data are manipulated repeatedly in an iterative fashion, the data will "return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for discrete state systems), their initial state" according to Wikipedia.  
     
  (Poincaré`s recurrence) Henri Poincaré, the great French mathematician, demonstrated that if one takes a set of data, and manipulate it, and then repeat that manipulation again and then again, at some point the original arrangement of the data will recur - Poincaré's recurrence! There is an illustration of this phenomena called "stretch and fold" where you take an image, let's say a picture of Poincaré, and you stretch it and fold it again and again so the picture is soon distorted into something like the static lines once seen on old TV sets, but at some point the original arrangement of the information recurs. In the stretch and fold demonstration of this mathematical phenomenon, illustrated here, the first partial return is at the 48th iteration, and the first full recurrence occurs at the 241nd iteration.
 
        
  Poincaré's recurrence shows us that, in fact, we should not be surprised that with serial dilution physiological effects disappear and return, (though with ultradilution the first recurrence is at the 15th to 18th iterations. Why the difference? I don't think there is much we can say about this yet, though we should note that one phenomenon, ultradilution, is a three dimensional manipulation of matter, and the stretch and fold is a "two dimensional" computer manipulation.  
   
 

"If a transformation is applied repeatedly to a mathematical system, and the system cannot leave a bounded region, it must return infinitely often to states near its original state."(1) Ian Stewart
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"fundamental"                iterated                 48th interaction                 iterations                    241 iteration
 
 
 





 
 

detailed discussions of Poincare's Recurrence can be found on line,
starting with wikipedia
and at Max Planck Institute for the Physics of Complex Systems

  
     
     
 

note: in his post http://paulbourke.net/fractals/stretch/, Paul Bourke see recurrence at the 188 iteration (and partial recurrence at the 98th iteration)

others (Max Planck Institute for the Physics of Complex Systems) see the return at 192 (and partial at 96)
and this source references an interesting example of recurrence at...
Arnold's Cat Map, which may be of interest here with reference to "recurrence"

 
     
 

 

 

 

  
     
  1. Stewart: Does God Play Dice: the Mathematics of Chaos